Mechanical signaling via nonlinear wavefront propagation in a mechanically excitable medium
Timon Idema1,2,*and Andrea J. Liu1,†
1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, USA 2Department of Bionanoscience, Kavli Institute of Nanoscience, Faculty of Applied Sciences,
Delft University of Technology, Delft, The Netherlands (Received 5 April 2013; published 23 June 2014)
Models that invoke nonlinear wavefront propagation in a chemically excitable medium are rife in the biological literature. Indeed, the idea that wavefront propagation can serve as a signaling mechanism has often been invoked to explain synchronization of developmental processes. In this paper we suggest a kind of signaling based not on diffusion of a chemical species but on the propagation of mechanical stress. We construct a theoretical approach to describe mechanical signaling as a nonlinear wavefront propagation problem and study its dependence on key variables such as the effective elasticity and damping of the medium.
DOI:10.1103/PhysRevE.89.062709 PACS number(s): 87.17.Aa, 87.53.Ay
I. INTRODUCTION
The physical phenomenon of nonlinear wavefront propaga-tion in an excitable medium is widely exploited by biological systems to transmit signals across many cells. For example, when the slime mold Dictyostelium begins to aggregate to form a fruiting body, wavefronts of the molecule cAMP propagate
across the amoeba colony [1–3]. Although cAMP itself spreads
diffusively, wavefronts of cAMP propagate ballistically across the amoeba colony because the colony is chemically excitable: when the local concentration of cAMP exceeds a threshold, further local release of the species is triggered [2,3]. Similar wavefronts of calcium and potassium, respectively, signal
fertilization in eggs [4] and the onset of spreading cortical
depression [5], associated with migraine auras.
Biological systems are typically overdamped, so they do not support sound waves and stress cannot propagate ballistically. In this paper we consider the possibility that mechanical signaling via ballistic propagation of a nonlinear wavefront can occur in a mechanically excitable medium, much as chemical signaling via ballistic propagation of a nonlinear wavefront can occur in a chemically excitable medium. In recent years there has been a growing recognition that mechanics plays an important role in biology, and that many cells sense and respond not only to chemical stimuli but also to mechanical stimuli [6–10]. This raises the possibility that mechanosensing at the cellular level could give rise to collective phenomena at larger length scales, such as collective
cell migration [11,12]. In this paper, we consider models
in which mechanosensing causes cells, or components of cells, to deform, thus generating more stress. We show that this nonlinear response to stress can lead to mechanically induced nonlinear wavefront propagation—a particular form of mechanical signaling—at the tissue level.
Several groups have previously studied waves in active
matter [11,13–22]. For instance, G¨unther and Kruse showed
how spontaneous oscillations in muscle fibers can lead to propagating waves in an overdamped viscoelastic medium
which by itself is linearly stable [15,16]. Both Banerjee and
*t.idema@tudelft.nl
†ajliu@physics.upenn.edu
Marchetti [17,18] and Radszuweit et al. [19] have studied the dynamics of viscoelastic polymer gels that can be contracted by molecular motors, and shown that the addition of motor dynamics to the overdamped system can lead to ballistic waves. Recently, K¨opf and Pismen showed that combining the properties of a polarizable, nonlinear elastic medium and chemo-mechanical coupling can lead to nonlinear instabilities as well, and be used to explain the properties of growing
epithelial tissue [20]. Building on this work, they also showed
that a similar model can lead to both long-wave and short-wave
instabilities and pattern formation [21]. In contrast to these
previous studies, where activity occurs continuously in space and time, we consider models of a different class of active matter, in which activity in the form of stress generation is localized to discrete positions in space and occurs only when activated by stress. We consider both overdamped elastic systems and viscoelastic systems, neither of which allow for the propagation of ballistic waves. Because we add activity only to discrete points, our approach allows us to directly relate the observable properties of the emerging waves to those of the underlying (visco)elastic medium.
It has recently been suggested that two very different biological systems might be mechanically excitable in this
sense: the early Drosophila embryo [23] and the developing
heart [24]. The early Drosophila embryo supports mitotic
wavefronts: nuclei at the poles of the embryo tend to divide first, giving rise to a mitotic wavefront separating dividing nuclei from those that have not yet divided. This
wavefront propagates across the entire embryo [25] at constant
speed [23]. Likewise, the heart tube of the avian embryo
beats via contractile wavefronts that are initiated at one end of the tissue and propagate across the heart tube with each
beat [26]. These two examples differ completely in their
biological details but are both described at a quantitative level
by models [23,24] that assume that mechanosensing leads to
the generation of more stress. In these systems, the active agents (the nuclei and cells) are discrete points which are separated by a passive medium. Therefore the existing models, which describe continuum active systems such as molecular motors distributed throughout a (visco)elastic gel, do not directly apply. It is therefore important to construct minimal models of mechanically excitable media and to understand their behavior more generally.
II. MODELS
We consider two minimal models of mechanically excitable media. In each case, stress can be generated at certain sites, or activatable nodes, if the local stress exceeds some threshold value. In the case of the early Drosophila embryo, these nodes would represent cell nuclei, while in the case of the developing heart tube, the nodes would represent cardiomyocytes (heart cells that can contract). In our models, the activatable nodes are embedded in a passive medium that supports mechanical stress and is overdamped. The nodes communicate with each other through this damped elastic medium: when a node is activated and stress is generated, it is transmitted through the medium, potentially causing further activation of other nodes. We solve these models and identify characteristic features exhibited by nonlinear wavefronts in such systems.
We start by considering two simple examples of the passive medium in which the activatable nodes are embedded. The elasticity of the medium is characterized by Lam´e coefficients λ and μ, or equivalently, by the Young’s modulus
Eand dimensionless Poisson ratio ν within linear elasticity
theory [27]. These parameters relate the stress σij inside the
elastic material to its strain (deformation) εij:
σij = E 1+ ν εij + ν 1− (d − 1)νεkkδij , (1) where d is the number of dimensions (2 or 3), and summation
over repeated indices is implied. The strain εij is defined in
terms of the displacement vector ui of the elastic material:
εij =12(∂jui+ ∂iuj). To avoid confusion we will follow the
usual convention and label the two-dimensional versions of the parameters with a subscript 2; they are related to
their three-dimensional counterparts by E2= E/(1 − ν2) and
ν2= ν/(1 − ν) [27]. The force per unit area is given by the
divergence of the stress: Pi = ∂jσij.
In the simplest model that we consider, corresponding to a thin elastic film that slides frictionally over a surface, we
balance the elastic force with a friction term ∂tui, where
is the friction coefficient. Such a system can be described by a two-dimensional model with the equation of motion:
∂tui = E2 2(1+ ν2) ∂j∂jui+ 1 1− ν2∂i∂juj . (2)
Note that Eq. (2), which describes the response of elastic
medium to a displacement ui, is similar to the diffusion
equation, but is a tensor equation instead of a scalar one. Mathematically, this model is the two-dimensional version of the overdamped limit of the viscoelastic gel model first introduced by Tanaka et al. [28].
The second model that we consider is a three-dimensional realization of an overdamped elastic medium, such as a polymer network immersed in a fluid. The elasticity of the
system is also described by Eq. (1), but the friction force is
proportional to the relative motion of the fluid and the elastic network: (∂tui− vi), where viis the fluid velocity. The stress
in an incompressible fluid depends linearly on the pressure p and the shear rate, ˙γijvisc=12(∂ivj + ∂jvi):
σijvisc= −pδij + 2η ˙γijvisc, (3)
in both two and three dimensions [29]. In the overdamped limit
(zero Reynolds number), taking the divergence of (3) gives the
Stokes equation. Combining the elastic and fluid equations gives a closed system for ui, vi, and p:
(∂tui− vi)= E 2(1+ ν) ∂j∂jui+ 1 1− (d − 1)ν∂i∂kuk , (4) (∂tui− vi)= ∂ip− η∂j∂jvi, (5) 0= ∂jvj. (6)
Equations (4)–(6) are identical to the two-fluid model studied
by Levine and Lubensky [30], but without the inertial terms.
We now add mechanical excitability as follows. We con-sider a collection of nodes at positions{ Rn}, where n indexes
the nodes. A node can be activated if some measure of the stress (for example, the absolute value of its largest eigenvalue) exceeds a threshold value α. If this occurs at time t, the node releases additional stress over a time interval t. For a node at Rnactivated at time t= tn, we therefore introduce an extra
force into Eq. (2), of the form
Piactive= ∂j[Qijδ(x − Rn) (t− tn) (tn+ t − t)]. (7)
Here Qij = fixj+ xifj is a symmetric tensor of rank 2,
corresponding to a force per unit volume f acting over a
distance x. Qij is therefore the symmetric combination of a
force and a distance, with the dimensions of a stress (force per unit area), so it represents a stress source. In two dimensions,
Qij = Q0δij + Q1(ninj −12δij) has an isotropic part of the
form Q0δij and a traceless anisotropic part of the form
Q1(ninj−12δij), where ˆnindicates the anisotropy direction.
The anisotropic contribution corresponds to a force dipole.
In three dimensions, Qij has a similar isotropic part and two
anisotropic parts that together span the plane perpendicular to the anisotropy direction. In general, we could also consider
asymmetric contributions to Qij, which would correspond to
torques (one in two dimensions and three in three dimensions). We solve for the response of the two-dimensional over-damped elastic medium of Eq. (2) to the active force in Eq. (7)
by deriving the Green’s tensor Gij k(x,t), which relates the
displacement uk(x,t) to a source term Qijδ(x) (t) at the origin
at time t = 0. We find that the material parameters E2, ν2, and
combine in two quantities with the dimensions of diffusion
constants, D1 = E2 1− ν2 2 = 2 1− ν2 μ , (8) D2 = E2 2(1+ ν2) = μ , (9)
which correspond to motion in the longitudinal and transverse directions, respectively, and together completely determine
the solution. Here μ= E2/2(1+ ν2)= E/2(1 + ν) is the
material’s shear modulus, which is the same in two and three
dimensions. The resulting Green’s tensor is given in Eq. (A1).
We can derive a similar solution for the response of the two-fluid model of Eqs. (4)–(6) to the active force in Eq. (7). In this case there is an extra parameter, the viscosity η of the fluid,
which gives rise to a natural relaxation timescale τ = μ/η of the system. In three dimensions, the three quantities governing the solution of Eqs. (4)–(6) are given by
D1= E 1− ν (1+ ν)(1 − 2ν) = 2(1− ν) 1− 2ν μ , (10) D2= E 1 2(1+ ν)= μ , (11) τ = E 2η(1+ ν) = μ η. (12)
The associated Green’s tensor is given by Eq. (A2).
III. RESULTS
Because the equations governing the passive medium are linear, we can now use the principle of superposition to study the effect of many source terms. We initialize the system
by activating a single node at the origin at t= 0. We then
measure the stress at the other nodes as a function of time, and activate them if they are above threshold by releasing more
stress, according to Eq. (7). We consider various cases for the
arrangement of the nodes: a regular triangular lattice, a random configuration with short-range correlations (as in a random packing of disks), and an uncorrelated random configuration. In addition, we look at variants in which the force term is purely isotropic (hydrostatic expansion/contraction) or is in the form of a volume-conserving force dipole, with either random orientation or orientations correlated to the direction of the traveling wavefront. In all cases, the model produces an activation wavefront with a well-defined speed, as shown
in Fig.1. We find that the speed of the wavefront depends on
the density of nodes but is insensitive to their arrangement. However, the spread of the wavefront around its mean
increases with the amount of randomness [Fig. 1(b)]. Not
surprisingly, if the orientations of the force dipoles are chosen
0.2 0.4 0.6 0.8 1.0
t
r
2 4 6 8 10 12 0 0.0 (a) 0 1 2 3 4 5 6 5 10 15 20v
0 (b)FIG. 1. (Color online) Calculated wavefronts. (a) Plots showing for each node (dots) the distance from the first activated node vs time of activation, with linear fits. This example has a traceless dipole source term (Q0= 0) and shows results for three different types of grids, all with the same density: regular triangular (gold), correlated random (blue), and uncorrelated random (red). (b) Mean wavefront speed for six realizations, with isotropic force term (1–3, blue, Q1= 0) and dipole force terms (4–6, red, Q0= 0), and on three different grids: regular triangular (circle, 1 and 4), correlated random (diamond, 2 and 5), and uncorrelated random (square, 3 and 6). Error bars indicate standard deviations. Inset shows the two-dimensional field of nodes, indicating the orientation in which nodes are activated, and color coded according to the time at which they are activated, on a hue scale (red-yellow-green-blue-violet).
at random, the speed of the wavefront is the same in all directions so that its shape is circular, as in the inset of Fig.1(b). In contrast, if the dipoles are all oriented in the same direction, the wavefront is no longer uniform but is faster in the direction of orientation. Also, for the same magnitude of the active force, the speed of the wavefront is somewhat higher if the force dipole Qij is isotropic (Q1= 0) than if it is a traceless force
dipole [Q0= 0, Fig.1(b)]. All these observations indicate that
the wavefront speed is dictated primarily by the properties of the medium and the average distance between nodes, and is insensitive to both the form of the active force and spatial arrangement of nodes.
Dimensional analysis of the material parameters of our
two-dimensional model [Eq. (2)] shows that there is only one
possible scaling for the wavefront speed with the material
parameters: v∼ E2/(a), where a is the grid spacing. The
dimensionless speed ¯v= (a/E2)v depends on the material’s
Poisson ratio ν2and the dimensionless threshold ¯α= αa2/Q,
where Q is the strength of the force term. We have determined the function ¯v(ν2,α¯) numerically for both isotropic and dipole force terms. We find that it obeys a fairly simple functional form, which can be motivated by an analytical argument based on the case of the simplest force term, the purely isotropic one (Q1= 0, so Qij = Q0δij). In this case, the stress is given by
σkliso= (1− ν2)Q0 x2 e −x2/4D1t × 2+ x 2 2D1t xkxl x2 + ν2 1− ν2 x2 2D1t − 1 δkl . (13) Because the stress drops off quadratically with distance, the major contribution to the stress at any node is due to forces exerted by neighboring nodes. Moreover, since the front expands radially, typically only a single nearest neighbor of any node will have been activated recently. We can therefore get a reasonable estimate for the local stress at a node by considering that nearest neighbor to be the only source. We
introduce the dimensionless time ¯t= E2t /(a2); then for a
single source a distance a away, the time at which the largest
eigenvalue of the stress reaches the dimensionless threshold ¯α
is given by ¯ α= (1 − ν2) 1+1+ ν2 2¯t e−(1−ν22)/4¯t. (14)
Unfortunately, Eq. (14) cannot be inverted analytically.
How-ever, the two factors containing ¯t are easily inverted, allowing us to make an educated guess for the functional form of the resulting dimensionless speed:
¯
v= −4(c1α¯+ c2) log( ¯α)
1− ν22 , (15)
where c1 and c2need to be determined numerically; we find
c1= 4.0 and c2= 1.5. As shown in Fig.2(a), the form given
by Eq. (15) works remarkably well. Moreover, the same
functional form also describes the results for a dipole force
term wavefront, as shown in Fig.2(b), the only difference being
the values of the two fit parameters—here we find c1= −1.0
0.2 0.4 0.6 0.8 1.0 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1.0 10 20 30 40 2
ν
2ν
v
(b)v
(a) 0 . 0 0 . 0FIG. 2. (Color online) Dimensionless wavefront speed as a func-tion of the Poisson ratio ν2and dimensionless threshold ¯α. Symbols indicate numerical solutions of the full system, lines the functional form of Eq. (15). (a) Isotropic force term Qij= Qδij. Fit parameters
c1= 4.0, c2= 1.5. Values of ¯α: 0.1 (blue/dots), 0.2 (red/ dia-monds), 0.3 (gold/squares) and 0.4 (green/triangles). (b) Dipole force term with random orientation angle θ : Qij = −Q cos(2θ)(δi1δj1− δi2δj2)− Q sin(2θ)(δi1δj2+ δi2δj1). Fit parameters c1= −1.0, c2= 1.0. Values of ¯α: 0.050 (blue/dots), 0.075 (red/diamonds), 0.100 (gold/squares), 0.125 (green/triangles up), 0.150 (purple/triangles down), and 0.200 (pink/hexagons).
In line with intuition, our model predicts that there is a
maximum threshold value ¯αmaxabove which a wavefront will
not propagate. This can happen for one of two reasons: either the force is not large enough to create a stress at the next node that exceeds the threshold value, or the nodes are so far apart that, due to the diffusive nature of the stress spreading, the threshold value is not reached. Both possibilities are contained
in the form of the dimensionless version of ¯αmax, given by
the maximum of the right-hand side of Eq. (14), which gives
¯
αmax= 2e−(1+ν2)/2 at ¯tmax= 12(1− ν2) and corresponding to
a minimum speed ¯vmin= 2/(1 − ν2). We note that as ν2
approaches its maximum value of 1, the minimum speed diverges, as can be seen in Fig.2(a).
For the three-dimensional two-fluid model of Eqs. (4)–(6),
there are two independent quantities with the dimensions of
speed, E/a and a/τ , where τ = η/μ is the material’s
relax-ation time [Eq. (12)]. We note that both of these scale linearly with the material’s Young’s modulus E (or equivalently, with the material’s shear modulus μ), which implies that also in this case the resulting wavefront velocity in a similar setup with excitable nodes will scale linearly with that modulus. It will
also scale with −nη1−n, where n is some number between 0
and 1, indicating that both the internal viscosity of the moving fluid and the friction between the elastic and viscous material contribute to the damping of the ballistic motion.
In both of our minimal models, the speed of the wavefront scales linearly with the elastic moduli. This distinguishes these wavefronts from ordinary elastic waves, where the speed of the wave scales as the square root of the elastic moduli, as in ordinary sound waves. We emphasize that this distinction stems from the fact that the wavefronts in our models are nonlinear phenomena, distinct from elastic waves.
IV. SUMMARY AND DISCUSSION
In this paper, we have introduced two theoretical realiza-tions of a mechanical signaling mechanism. We have shown that nonlinear wavefront propagation is a robust feature of both models. In both cases, the wavefront velocity is insensitive to
the spatial distribution of excitable nodes. It is also insensitive to whether the stress is released in an isotropic or traceless anisotropic fashion. Furthermore, a fundamental feature of both models is that the wavefront velocity is proportional to the Young’s modulus of the medium, and the magnitude of the velocity can be understood simply and quantitatively in terms of characteristic dimensionless variables, such as the stress threshold made dimensionless with the magnitude of the force dipole released when a node is excited. We note that the linear scaling of the wavefront propagation speed with the medium’s Young’s modulus is a direct consequence of the fact that in our model the wave propagates through a passive, mechanical medium, in contrast to propagation through an active gel.
The overdamped elastic models considered here are the simplest models that could be used to describe a tissue. It would be worthwhile to explore mechanical signaling in other models that have been proposed for tissues, including the active gel model [11,13–21] and cellular models [31,32]. The active gel
model first proposed by Kruse et al. [13,14] is an extension
of the two-fluid model of Levine and Lubensky [30] with
a continually active (energy-consuming) term to model the dynamics of the cytoskeleton due to motor activity. As we have shown here, such a continuous activity is not necessary to describe wavefront propagation, as local and discrete activity is sufficient. However, given the presence of active motors in the cytoskeleton, it would be interesting to see how the wavefront is affected by an active term in the model. It would also be interesting to compare the results of such an interaction with those of Bois et al., who study pattern formation in active
fluids due to chemical signaling [11]. These active models,
and the one we used here, are continuous models. However, tissues are, of course, composed of discrete units, the cells. As shown by Manning et al., several mechanical properties of the tissue, such as its surface tension, are determined by cell-cell
adhesion and cortical tension [31]. Recent work by Chiou et al.
provides a method to measure the relative magnitude of forces
acting within and between cells [32]. These results now make
it possible to construct a quantitative cell-based tissue model in which wavefront propagation due to mechanical signaling can be studied as well.
Now that we have introduced a minimal model for me-chanical signaling via nonlinear wavefront propagation, we can ask how one might identify biological contexts in which mechanical signaling is likely to occur. We can also ask how to determine whether a given wavefront is an example of mechanical signaling. Wavefronts of processes that generate stresses are obvious likely candidates. In order for a medium to be mechanically excitable, however, it is not enough to have a collection of nodes capable of generating stress. There must also be a mechanism to sense when a stress threshold is reached. Several such mechanisms have been
identified [33], including stress-dependent ion channels that
can release ions above a threshold stress [34–37], cell-cell
adhesion complexes such as cadherin complexes [38], focal
adhesions [39], and integrins [40]. Contractile wavefronts in
heart tissue represent a good candidate for mechanical
signal-ing [24]; cardiomyocytes generate stress as they contract via
the excitation-contraction mechanism [41,42], and they sense
stress as well—the contraction amplitude of a cardiomyocyte on a gel depends sensitively on the stiffness of the gel [24,43].
In addition, it is known that mechanical palpitation can disrupt
contractile wavefronts [44]. Another possible realization is
spreading cortical depression, which involves a potassium
wavefront [5] that can be triggered mechanically [45]. These
examples suggest that it may be worthwhile to reexamine nonlinear wavefronts in biological contexts to see if they are more properly interpreted as mechanical signaling.
ACKNOWLEDGMENTS
We thank Gareth Alexander, Michael Lampson, Tom Lubensky, and Phil Nelson for instructive discussions. This work was partially supported by the Netherlands Organization for Scientific Research through a Rubicon grant (T.I.) and by NSF-DMR-1104637 (T.I. and A.J.L.).
APPENDIX: EXPRESSIONS FOR THE GREEN’S TENSORS 1. Two-dimensional elastic model
In the simplest model that we consider, corresponding to a thin elastic film that slides frictionally over a surface, we balance
the elastic force ∂jσij with a friction term ∂tui, where is the friction coefficient. Such a system can be described by a
two-dimensional model with an equation of motion given by Eq. (2). The two-dimensional Green’s tensor Gij k(x,t) associated
with Eq. (2), relating the tensor Qij in a source term ∂jQijδ(x) (t) to an output uk, is given by
Gij k(x,t) = − 1 μx 1− ν2 2 + 8D2t x2 e−x2/4D1t− 1+8D2t x2 e−x2/4D2t xixjxk x3 −2D2t x2 [e −x2/4D1t − e−x2/4D2t ]φij k+ e−x 2/4D2t δik xj x , (A1) where μ= E2/2(1+ ν2), D1= (2/(1 − ν2))(μ/ ), D2= μ/ , x =√x · x, and φij k= δijxxk + δik xj x + δj k xi x.
2. Three-dimensional two-fluid model
In the two-fluid model, we couple an elastic mesh, described by a displacement field ui, to an incompressible viscous fluid,
described by a velocity field vi. The resulting equations are Eqs. (4)–(6), and the associated Green’s tensor is given by
Gij k(x,t) = Ghomij k (x,t) + Gstatij k(x,t), (A2)
Ghomij k (x,t) = − 1 (2π x)2D 1 A D1t x2 xixjxk x3 − B D1t x2 φij k + e−t/τ (2π x)2D 2 A D2t x2 xixjxk x3 − B D2t x2 φij k − e−t/τ (2π x)2D 2 C D2t x2 xj x δik, (A3) Gstatij k(x,t) = −2π (1+ ν) (2π x)2E δik xj x − 3(1+ ν) 8π x2E(1− ν) xixjxk x3 + (1+ ν) 8π x2E(1− ν)φij k (A4)
where μ= E/2(1 + ν), D1= ((2 − 2ν)/(1 − 2ν))(μ/ ), D2= μ/ , τ = μ/η, and
A(y)= 15√πy+ π y e−1/4y+ 3 2 − 15y πerf 1 2√y , (A5)
B(y)= 3√πye−1/4y+
1 2− 3y πerf 1 2√y , (A6) C(y)= π ye −1/4y− πerf 1 2√y . (A7)
For an input term that runs only over a time interval t as in Eq. (7) of the main text, continuity demands that for t < t we
[1] P. Devreotes,Science 245,1054(1989).
[2] H. Levine and W. Reynolds,Phys. Rev. Lett. 66,2400(1991). [3] K. J. Lee, E. C. Cox, and R. E. Goldstein,Phys. Rev. Lett. 76,
1174(1996).
[4] J. D. Lechleiter and D. E. Clapham,Cell 69,283(1992). [5] N. A. Goroleva and J. Bures,J. Neurobiol. 14,353(1983). [6] G. H. Altman, R. L. Horan, I. Martin, J. Farhadi, P. R. H. Stark,
V. Volloch, J. C. Richmond, G. Vunjak-Novakovic, and D. L. Kaplan,FASEB J. 16,270(2001).
[7] D. E. Discher, P. Janmey, and Yu-li Wang,Science 310,1139 (2005).
[8] V. Vogel and M. Sheetz, Nat. Rev. Mol. Cell Biol. 7, 265 (2006).
[9] D. E. Discher, D. J. Mooney, and P. W. Zandstra,Science 324, 1673(2009).
[10] S. D. Subramony, B. R. Dargis, M. Castillo, E. U. Azeloglu, M. S. Tracey, A. Su, and H. H. Lu,Biomaterials 34,1942(2013). [11] J. S. Bois, F. J¨ulicher, and S. W. Grill,Phys. Rev. Lett. 106,
028103(2011).
[12] X. Serra-Picamal, V. Conte, R. Vincent, E. Anon, D. T. Tambe, E. Bazellieres, J. P. Butler, J. J. Fredberg, and X. Trepat,Nat. Phys. 8,628(2012).
[13] K. Kruse, J.-F. Joanny, F. J¨ulicher, J. Prost, and K. Sekimoto, Phys. Rev. Lett. 92,078101(2004).
[14] K. Kruse, J.-F. Joanny, F. J¨ulicher, J. Prost, and K. Sekimoto, Eur. Phys. J. E 16,5(2005).
[15] S. G¨unther and K. Kruse,New J. Phys. 9,417(2007). [16] S. G¨unther and K. Kruse,Chaos 20,045122(2010). [17] S. Banerjee and M. C. Marchetti,Soft Matter 7,463(2011). [18] S. Banerjee, T. B. Liverpool, and M. C. Marchetti,Europhys.
Lett. 96,58004(2011).
[19] M. Radszuweit, S. Alonso, H. Engel, and M. B¨ar,Phys. Rev. Lett. 110,138102(2013).
[20] M. H. K¨opf and L. M. Pismen,Soft Matter 9,3727(2013). [21] M. H. K¨opf and L. M. Pismen,Physica D 259,48(2013). [22] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool,
J. Prost, Madan Rao, and R. Aditi Simha,Rev. Mod. Phys. 85, 1143(2013).
[23] T. Idema, J. O. Dubuis, L. Kang, M. L. Manning, P. C. Nelson, T. C. Lubensky, and A. J. Liu,PLoS ONE 8,e77216 (2013).
[24] S. Majkut, T. Idema, J. Swift, C. Krieger, A. J. Liu, and D. E. Discher,Curr. Biol. 23,2434(2013).
[25] V. E. Foe and B. M. Alberts, J. Cell Sci. 61, 31 (1983). [26] F. de Jong, T. Opthof, A. A. Wilde, M. J. Janse, R. Charles,
W. H. Lamers, and A. F. Moorman,Circ. Res. 71,240(1992). [27] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed.
(Butterworth Heinemann, Burlington, MA, 1986).
[28] T. Tanaka, L. O. Hocker, and G. B. Benedek,J. Chem. Phys. 59, 5151(1973).
[29] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Butterworth Heinemann, Burlington, MA, 1987).
[30] A. J. Levine and T. C. Lubensky, Phys. Rev. E 63, 041510 (2001).
[31] M. L. Manning, R. A. Foty, M. S. Steinberg, and E.-M. Schoetz, Proc. Nat. Acad. Sci. USA 107,12517(2010).
[32] K. K. Chiou, L. Hufnagel, and B. I. Shraiman,PLoS Comput. Biol. 8,e1002512(2012).
[33] C. C. DuFort, M. J. Paszek, and V. M. Weaver,Nat. Rev. Mol. Cell Biol. 12,308(2011).
[34] P. G. Gillespie and R. G. Walker,Nature (London) 413, 194 (2001).
[35] C. Kung,Nature (London) 436,647(2005).
[36] C. Kung, B. Martinac, and S. Sukharev,Annu. Rev. Microbiol. 64,313(2010).
[37] S. Sukharev and F. Sachs,J. Cell Sci. 125,3075(2012). [38] E. Tzima, M. Irani-Tehrani, W. B. Kiosses, E. Dejana, D. A.
Schultz, B. Engelhardt, G. Cao, H. DeLisser, and M. A. Schwartz,Nature (London) 437,426(2005).
[39] B. Geiger, J. P. Spatz, and A. D. Bershadsky,Nat. Rev. Mol. Cell Biol. 10,21(2009).
[40] A. van der Flier and A. Sonnenberg,Cell Tissue Res. 305,285 (2001).
[41] H. E. Huxley,Science 164,1356(1969).
[42] C. C. Ashley, I. P. Mulligan, and T. J. Lea,Q. Rev. Biophys. 24, 1(2009).
[43] A. J. Engler, S. Sen, H. L. Sweeney, and D. E. Discher,Cell 126,677(2006).
[44] M. R. Franz, R. Cima, D. Wang, D. Profitt, and R. Kurz, Circulation 86,968(1992).
[45] S. Akerman, P. R. Holland, and P. J. Goadsby,Brain Res. 1229, 27(2008).
